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%I #2 Mar 30 2012 18:37:21
%S 1,1,1,10,4,1,447,72,9,1,50040,4624,264,16,1,10435970,683300,23750,
%T 700,25,1,3470932404,178979256,4569480,84840,1530,36,1,1677020809366,
%U 72215891104,1489987002,20776980,241325,2940,49,1,1106343610197376
%N Triangle T where the g.f. of row n of T^(2n) = (2n^2 + y)^n for n>=0, as read by rows, where T^n denotes the n-th matrix power of T.
%C Analogous to Pascal's triangle, C, which obeys a similar rule: the g.f. of row n of C^(2n) = (2n + y)^n for n>=0.
%C Conjecture: for all integer k, there exists an integer triangle J such that the g.f. of row n of J^(k*n) = (k*n^2 + y)^n for n>=0.
%e Triangle T begins:
%e 1;
%e 1,1;
%e 10,4,1;
%e 447,72,9,1;
%e 50040,4624,264,16,1;
%e 10435970,683300,23750,700,25,1;
%e 3470932404,178979256,4569480,84840,1530,36,1;
%e 1677020809366,72215891104,1489987002,20776980,241325,2940,49,1;
%e 1106343610197376,41253720775296,725138126272,8309193088,73585120,586432,5152,64,1;
%e 953498812570622640,31544658525648240,487943071058088,4827635270640,35544216204,218340360,1269324,8424,81,1;
%e ...
%e Matrix square T^2 begins:
%e 1;
%e 2,1; <== (2 + y)^1 = g.f. for row 1 of T^2
%e 24,8,1;
%e 1056,180,18,1;
%e 114496,11456,672,32,1;
%e 23356640,1627600,60400,1800,50,1;
%e ...
%e Matrix power T^4 begins:
%e 1;
%e 4,1;
%e 64,16,1; <== (2*2^2 + y)^2 = g.f. for row 2 of T^4
%e 2904,504,36,1;
%e 301824,34048,1920,64,1;
%e 59043680,4635200,186800,5200,100,1;
%e ...
%e Matrix power T^6 begins:
%e 1;
%e 6,1;
%e 120,24,1;
%e 5832,972,54,1; <== (2*3^2 + y)^3 = g.f. for row 3 of T^6
%e 598080,72384,3744,96,1;
%e 113094720,9838800,408000,10200,150,1;
%e ...
%e Matrix power T^8 begins:
%e 1;
%e 8,1;
%e 192,32,1;
%e 10128,1584,72,1;
%e 1048576,131072,6144,128,1; <== (2*4^2 + y)^4 = g.f. for row 4 of T^8
%e 193866560,18284800,752800,16800,200,1;
%e ...
%o (PARI) {T(n,k,p=2)=local(M=Mat(1), N, L);for(i=1, n, N=M; M=matrix(#N+1, #N+1, r, c, if(r>=c, if(r<=#N, (N^(p*(#N)))[r, c], polcoeff((x+p*(#M)^2)^(#M), c-1)))); L=sum(i=1, #M, -(M^0-M)^i/i);M=sum(i=0, #M, (L/p/(#N))^i/i!); ); M[n+1, k+1]}
%Y Cf. columns: A177391, A177392, A177393, row sums: A177394.
%Y Cf. variant: A132870.
%K nonn,tabl
%O 0,4
%A _Paul D. Hanna_, May 25 2010